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Chapter 3: Problem 20

Construct a truth table for the given statement. \(\sim(q \leftrightarrow p)\)

Short Answer

Expert verified
The truth values of \( \sim(q \leftrightarrow p) \) are: for TT it's F, TF it's T, FT it's T, FF it's F.

Step by step solution

01

Specify every possible true-false value of \(p\) and \(q\)

Create two columns for the variables \(p\) and \(q\). Since there are two variables, there are \(2^2 = 4\) possible combinations of true (T) and false (F): TT, TF, FT, FF.
02

Calculate the truth value of \(q \leftrightarrow p\)

The biconditional operator \( \leftrightarrow \) is true if both \(p\) and \(q\) have the same truth value, i.e. both are true or both are false. Create a third column for \(q \leftrightarrow p\) and fill it based on the logic: if both \(p\) and \(q\) have the same truth value, put true, otherwise put false.
03

Calculate the truth value of \(\sim(q \leftrightarrow p)\)

The NOT operator \( \sim \) is true if the operand is false, and vice versa. Create a fourth column for \(\sim(q \leftrightarrow p)\) and fill it based on this logic: if \(q \leftrightarrow p\) is true, put false, and if it's false, put true.

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